Quasi-concave functions on meet-semilattices

This paper deals with maximization of set f'unctions delined as minimum values of monotone linkage functions. In previous research, it has been shown that such a set function can be maximized by a greedy type algorithm over a family of all subsets of a finite set. ln this paper, we extend this finding to meet-semilattices. We show that the class of functions defined as minimum values of monotone linkage functions coincides with the class of quasiconcave set functions. Quasi-concave functions determine a chain of upper level sets each of which is a meet-semilattice. This structure allows development of a polynomial algorithm that finds a minimal set on which the value of a quasi-concave function is maximum. One of the critical steps of this algorithm is a set closure. Some examples of closure computation, in particular, a closure operator for convex geometries, are considercd. @ 2007 Elsevier B.V. All richts reserved. Kelwords: Closure operator; Greedy algorithm; Monotone linkage function; Quasi-concave function; Semilattice